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%I #32 Sep 08 2022 08:44:50
%S 1,0,0,2,2,2,0,0,0,2,0,0,4,0,0,2,2,0,0,0,4,0,0,2,0,2,0,2,0,0,0,2,0,2,
%T 0,0,6,2,0,0,0,0,0,0,2,4,0,0,4,0,0,0,0,0,0,2,0,0,0,2,8,0,0,0,2,0,0,2,
%U 0,2,0,2,0,0,0,4,0,0,0,0,4,4,0,0,0,0,0,0,0,2,0,0,4,2,0,0,0,2,0,2,6,0,0,0,0
%N Theta series of quadratic form (or lattice) with Gram matrix [ 3, 1; 1, 4 ].
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C The number of integer solutions (x, y) to 3*x^2 + 2*x*y + 4*y^2, discriminant -44. - _Ray Chandler_, Jul 12 2014
%H John Cannon, <a href="/A028953/b028953.txt">Table of n, a(n) for n = 0..10000</a>
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of phi(q) * phi(q^11) - 2*q * f(-q^2) * f(-q^22) = phi(q^3) * phi(q^33) + 2*q^4 * chi(q) * psi(-q^3) * chi(q^11) * psi(-q^33) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - _Michael Somos_ and Alex Berkovich, Jun 24 2011
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (44 t)) = 44^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Jun 24 2011
%F G.f.: Sum_{n, m in Z} x ^ (3*n*n + 2*n*m + 4*m*m).
%F a(4*n + 2) = a(11*n + 2) = a(11*n + 6) = a(11*n + 7) = a(11*n + 8) = a(11*n + 10) = 0. - _Michael Somos_, Feb 23 2012
%e G.f. = 1 + 2*q^3 + 2*q^4 + 2*q^5 + 2*q^9 + 4*q^12 + 2*q^15 + 2*q^16 + 4*q^20 + 2*q^23 + 2*q^25 + 2*q^27 + 2*q^31 + 2*q^33 + 6*q^36 + 2*q^37 + 2*q^44 + 4*q^45 + ...
%t r[n_] := Reduce[{x, y}.{{3, 1}, {1, 4}}.{x, y} == n, {x, y}, Integers]; Table[rn = r[n]; Which[rn === False, 0, Head[rn] === Or, Length[rn], Head[rn] === And, 1], {n, 0, 105}] (* _Jean-François Alcover_, Nov 05 2015 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^11] - 2 q QPochhammer[ q^2] QPochhammer[ q^22], {q, 0, n}]; (* _Michael Somos_, Feb 09 2017 *)
%o (PARI) {a(n) = if( n<1, n==0, qfrep([3, 1; 1, 4], n)[n] * 2)}; /* _Michael Somos_, Jun 24 2011 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=1, sqrtint( n), 2 * x^k^2, 1 + A) * sum( k=1, sqrtint( n\11), 2 * x^(11*k^2), 1 + A) - 2 * x * eta(x^2 + A) * eta(x^22 + A), n))}; /* _Michael Somos_, Jun 24 2011 */
%o (Magma) A := Basis( ModularForms( Gamma1(44), 1), 87); A[1] + 2*A[4] + 2*A[5] + 2*A[6] + 2*A[10] + 4*A[13] + 2*A[16] + 2*A[17] + 4*A[21] + 2*A[24]; /* _Michael Somos_, Feb 09 2017 */
%Y Cf. A028952, A106282.
%K nonn
%O 0,4
%A _N. J. A. Sloane_.
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